 Solution

The fan of the affine line consists of the zero cone and ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. Under $\phi (e_1)=0$ and $\phi (e_2)=e^{\prime }$, the four primitive source rays map asEvery cone of the source fan consequently maps into ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. The defining criterion for a toric morphism is satisfied, so $\phi$ inducesUnder the blowup description from part (i), this is the blowup map followed by the projection ${\mathbb{P}}^1\times {\mathbb{A}}^1\to {\mathbb{A}}^1$.